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Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties

Fabio V. Difonzo

Abstract

Letting $P$ be a convex polytope in $\mathbb{R}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in $\mathbb{R}^n$ with at most $n-d-1$ vertices, and provide results about continuity and differentiability for the corresponding set-valued maps.

Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties

Abstract

Letting be a convex polytope in with vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point . We prove that such sets are polytopes in with at most vertices, and provide results about continuity and differentiability for the corresponding set-valued maps.
Paper Structure (5 sections, 7 theorems, 39 equations)

This paper contains 5 sections, 7 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Geometric structure of $\Lambda(p)$ and $\Gamma(p)$
  4. Analytical Properties of $\Lambda(p)$ and $\Gamma(p)$
  5. Conclusions

Key Result

Proposition 3.1

For each $p\in P$, $\Gamma(p)$ and $\Lambda(p)$ are polytopes of dimension at most $n-d-1$.

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 3.5
  • Example 3.6
  • ...and 10 more